Introduction to the Renormalization group method as a powerful reduction method of dynamics

Extracting effective slow dynamics with fewer degrees of freedom from a complex system with many degrees of freedom is of basic importance in all areas of Science. Typical examples include the derivation of the amplitude and phase dynamics from nonlinear oscillators, that of the Boltzmann equation from Hamilton dynamics, which is further reduced to fluid dynamics and so on. The purpose of this talk is to give an elementary introduction to the renormalization group (RG) method as a powerful reduction method of differential (difference) equations in terms of the notion of envelopes. Some simple examples will be worked out in this method, which include the van der Pol (Rayleigh) equation with its discrete analog and a generic system with a bifurcation. In the final part, we list up various examples to which the RG method has been successfully applied.